| L(s) = 1 | − 3·3-s + 14·5-s + 24·7-s + 9·9-s + 28·11-s − 74·13-s − 42·15-s + 82·17-s − 92·19-s − 72·21-s − 8·23-s + 71·25-s − 27·27-s − 138·29-s − 80·31-s − 84·33-s + 336·35-s + 30·37-s + 222·39-s + 282·41-s − 4·43-s + 126·45-s − 240·47-s + 233·49-s − 246·51-s − 130·53-s + 392·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.25·5-s + 1.29·7-s + 1/3·9-s + 0.767·11-s − 1.57·13-s − 0.722·15-s + 1.16·17-s − 1.11·19-s − 0.748·21-s − 0.0725·23-s + 0.567·25-s − 0.192·27-s − 0.883·29-s − 0.463·31-s − 0.443·33-s + 1.62·35-s + 0.133·37-s + 0.911·39-s + 1.07·41-s − 0.0141·43-s + 0.417·45-s − 0.744·47-s + 0.679·49-s − 0.675·51-s − 0.336·53-s + 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.456475139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456475139\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 82 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 T + p^{3} T^{2} \) |
| 29 | \( 1 + 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 30 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 130 T + p^{3} T^{2} \) |
| 59 | \( 1 + 596 T + p^{3} T^{2} \) |
| 61 | \( 1 + 218 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 856 T + p^{3} T^{2} \) |
| 73 | \( 1 + 998 T + p^{3} T^{2} \) |
| 79 | \( 1 - 32 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 + 246 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.77330049891398070825551359400, −14.29044469741827835703743889088, −12.76380976069856879883216347526, −11.66210973436214374273958040520, −10.39857893157220476481864167577, −9.317060492462773161889796243369, −7.57312080829463688501265645696, −5.96338549533946610740806585040, −4.77987215517671433999133392516, −1.82647386098955805979520473112,
1.82647386098955805979520473112, 4.77987215517671433999133392516, 5.96338549533946610740806585040, 7.57312080829463688501265645696, 9.317060492462773161889796243369, 10.39857893157220476481864167577, 11.66210973436214374273958040520, 12.76380976069856879883216347526, 14.29044469741827835703743889088, 14.77330049891398070825551359400
